Towards a L 2 cohomology theory for Hodge modules on infinite covering spaces: L 2 constructible cohomology and L 2 de Rham cohomology for coherent D-modules

Abstract

This article constructs Von Neumann invariants for constructible complexes and coherent D-modules on compact complex manifolds, generalizing the work of the author on coherent L 2 -cohomology. We formulate a conjectural generalization of Dingoyan's L 2 -Mixed Hodge structures in terms of Saito's Mixed Hodge Modules and give partial results in this direction.